Vektorska polja na sferah : delo diplomskega seminarja

Baltič, Mark (author), Vavpetič, Aleš (mentor)

V diplomskem delu si ogledamo nekaj dejstev o obstoju neničelnega zveznega tangentnega vektorska polja na $n$-dimenzionalnih sferah. Najprej eksplicitno pokažemo konstrukcijo oz. nakažemo neobstoj takega vektorskega polja pri nižjih dimenzijah, nato pa idejo posplošimo na višje dimenzije. Pri dveh dimenzijah si ogledamo tudi druge objekte in nakažemo indikator, ki določi število izoliranih točk, kjer vektorsko polje izgine. Kot posledico glavnega izreka dokažemo tudi Browerjev izrek o negibni točki.

• Language: Slovenian
• Keywords: tangentna vektorska polja, sfera, Browerjev izrek o negibni točki, Eulerjeva karakteristika
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-101338
• UDC: 515.1
• COBISS.SI-ID: 18394969
• Publication date in RUL: 30.05.2018

Secondary language

• Language: English
• Title: Hairy ball theorem

Abstract

In this thesis we show some facts about existence of a non vanishing continuous tangent vector field on a $n$-dimensional sphere. At first we explicitly construct such vector field or we indicate the non existence of such vector field at low dimensions and then we generalize the idea to higher dimensions. In the two-dimensional case we also examine other objects and point out the indicator that sets the number of isolated points at which the vector field vanishes. As a consequence of our main theorem we also prove the Brower fixed-point theorem.

• Keywords: tangent vector fields, sphere, Brouwer's fixed-point theorem, Euler characteristic